We kindly ask all participants, including remote ones and those
watching recorded videos, to register
at this link.
A course of 10-12 lectures will be devoted to the main issues of the theory
of optimal control: the existence of solutions and the necessary optimality
conditions, with a presentation of the required mathematical foundation. The
course focuses on two major results of the theory:
— Sufficient Filippov conditions for the existence of an optimal control;
— Pontryagin's maximum principle.
Both results will be presented and proved in an invariant geometric form
for problems on smooth manifolds.
Along the way, important general mathematical material will be presented:
— Measurable sets and functions, Carathéodory differential equations;
— Differential equations on smooth manifolds;
— Elements of chronological calculation R.V. Gamkrelidze-A.A. Agracheva;
— Differential forms;
— Elements of symplectic geometry.
The course is designed for students of mathematics and physics (starting
from the 3rd year) and graduate students. No prior knowledge of control theory
is assumed.
The course is a continuation and deepening of the online course given by
Yu.L. Sachkov in 2022, but it can also be listened to independently. At the end of
the course, an exam is planned for those who wish.
Course program
- Statement of the optimal control problem.
- Measurable sets and functions, Carathéodory differential equations.
- Sufficient Filippov conditions for the existence of an optimal control.
- Differential equations on smooth manifolds.
- Elements of chronological calculus R.V. Gamkrelidze-A.A. Agrachev.
- Differential forms.
- Elements of symplectic geometry.
- Proof of the Pontryagin maximum principle on manifolds: geometric form,
optimal control problems with different boundary conditions.
References
[1] A.A. Agrachev, Yu.L. Sachkov, Geometric control theory. M.: Fizmatlit, 2005. Engl. trans.: A.A. Agrachev, Yu.L. Sachkov, Control theory from the geometric viewpoint. Springer-Verlag.
2004
[2] A.A. Agrachev, D. Barilari, U. Boscain, A Comprehensive Introduction to sub-Riemannian
Geometry from Hamiltonian viewpoint. Cambr. Stud. in Advanced Math., Cambr. Univ. Press,
2019.
[3] Yu.L. Sachkov, Introduction to geometric control theory. M.: URSS, 2021, 160 pp. Engl.
trans.: Yu. Sachkov, Introduction to geometric control. Springer Verlag, 2022.
 RSS: Forthcoming seminars
 Lecturer
Sachkov Yurii Leonidovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
